Sieve Of Euler

February 25, 2011

The ancient Sieve of Eratosthenes that computes the list of prime numbers is inefficient in the sense that some composite numbers are struck out more than once; for instance, 21 is struck out by both 3 and 7. The great Swiss mathematician Leonhard Euler invented a sieve that strikes out each composite number exactly once, at the cost of some additional bookkeeping. It works like this:

First, make a list of numbers from 2, as large as you wish; call the maximum number n.

Second, extract the first number from the list, make a new list in which each element of the original list, including the first, is multiplied by the extracted first number.

Third, “subtract” the new list from the original, keeping in an output list only those numbers in the original list that do not appear in the new list.

Fourth, output the first number from the list, which is prime, and repeat the second, third and fourth steps on the reduced list excluding its first element, continuing until the input list is exhausted.

Just as in the Sieve of Eratosthenes, you can speed up the Sieve of Euler by considering only odd numbers, by stopping once the first item in the list is greater than the square root of n, and by computing the new list in the second step only as far as n.

Your task is to write a program that implements the Sieve of Euler, then compare its performance to the Sieve of Eratosthenes. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

Strictly speaking, it doesn’t build the second list and subtract—the best way I could find to do this was
with sets, but the time complexity was still larger than it neede to be—but it does filter out all multiples
of the first item once and only once. I tried to get a faster version going using iterators (kind of like lazy
lists), but it turned out to be slower. Eratosthenes still wins; see codepad for timing, where I
pit this against the Python Cookbook’s erat2() implementation.